Friday, December 03, 2010

Completing Zeno

(1) At time t2 Achilles reaches a point where the tortoise was at t1, at t3 he reaches the point where the tortoise was at t2, at t4 he reaches the point where the tortoise was at t3, and so on.

(2) Achilles will not reach the tortoise before the sequence outlined in (1) above is completed,

(3) The sequence is never completed.

(4) Achilles will never reach the tortoise.

Unlike the IEP version I referenced below, the conclusion appears to follow logically from the premisses. Furthermore, it does not rely on assumptions like 'cannot do infinitely many things in a finite time'. It relies simply on the definition of an infinite sequence as one which is endless, not terminated, not completed.

Well it appears valid, but is it? More tomorrow, and comments welcome.

3 Comments:

This, I think, is more convincing. But what is meant by (3)? If the claim is '~∃t. ∀n t>tn' then that too is false. The abstract presentation of (1) might suggest (3) because it's easy to imagine the ts wandering off to infinity. But a concrete example, say Tortoise starts 1 ahead of Achilles, Achilles runs at speed 1, Tortoise crawls at speed 1/2, soon corrects this, as the ts turn out to be (0, 1, 3/2, 7/4, 15/8,...}, with tn=2-2/2^n.

My guess is that the confusion arises because we tend to think of infinite sequences by means of spatial or temporal metaphor. Take the squares. We imagine counting 0, 1, 4, 9, 16,..., silently in our heads, one at a time. The metaphor gets confused with the subject when the subject is itself temporal. We think of the times 0, 1, 3/2, 7/4,... 'occuring' at times 0, 1, 2, 3,... The confusion vanishes when we think of a sequence as a function from the natural numbers. OK, we have to accept an actually infinite domain, but there is no sense in which the function is 'incomplete'. There is a notion of incompleteness of functions but that's the concept of a partial function, which doesn't apply in this case since there's a well defined tn for every n.

Brandon seems to be trying to get a lot of meta-philosophical mileage out of this paradox. What do you make of that?

As you have spotted, there is something seriously wrong with the argument above, but I shall remain silent until tomorrow.

Brandon’s point (I think) is that the ‘Standard Solution’ rejects the assumptions of the paradox, and makes a distinction accordingly, but he objects that this is not ‘solving a paradox’, since this requires also showing good reason for rejecting the assumptions. I stepped back from this and questioned whether the formulation he was using had any paradoxical flavour at all, and we left it there. I certainly don’t agree with him on the idea of ‘apparent inconsistency’ and the notion of ‘apparent validity’ which it implies. As you know from our previous engagements and skirmishes, I believe that any ‘apparent inconsistency’ should be teased out until it becomes a real inconsistency, and we do this by re-formulating the argument with care. Or rather (since there can be no ‘real inconsistency’) we repeat the conjuring trick by showing it in greater detail, until the sleight-of-hand becomes obvious.